Binomial Theorem

Question
CBSEENMA11014378

How many litres of water will have to be added to 1125 litres of the 45%, solution of acid so that the resulting mixture will contain more than 25% but less than 30% acid content?

Solution

Quantity of 45% acid solution = 1125 litres
Quantity of acid = fraction numerator 1125 cross times 45 over denominator 100 end fraction litres space equals space 2025 over 4 space litres
Suppose x litres of water is added
∴          Quantity of acid = 2025 over 4 space litres
Total quantity of mixture = (1125 + x) litres
             % of acid content = space space fraction numerator 2025 over denominator 4 space left parenthesis 1125 plus straight x right parenthesis end fraction cross times 100
According to the problem,
                     space space space space space 25 less or equal than fraction numerator 2025 over denominator 4 space left parenthesis 1125 plus straight x right parenthesis end fraction cross times 100 less or equal than 30
rightwards double arrow                     25 less or equal than fraction numerator 50625 over denominator 1125 plus straight x end fraction less or equal than 30
Now,                 25 less or equal than fraction numerator 50625 over denominator 1125 plus straight x end fraction space space rightwards double arrow space space 1 less or equal than fraction numerator 2025 over denominator 1125 plus straight x end fraction space rightwards double arrow space 1125 plus straight x space less or equal than space 2025 space rightwards double arrow space straight x space less or equal than space 900     ...(i)
Also,        fraction numerator 50625 over denominator 1125 plus straight x end fraction less or equal than 30 space rightwards double arrow space fraction numerator 10125 over denominator 1125 plus straight x end fraction less or equal than 6 space space rightwards double arrow space 10125 space less or equal than space 6750 space plus space 6 straight x space rightwards double arrow space 3375 less or equal than 6 straight x
rightwards double arrow                         6 straight x greater or equal than 3375
rightwards double arrow                          2 straight x greater or equal than 1125 space rightwards double arrow space straight x greater or equal than 562.5                                                                        ...(ii)
From (i) and (ii), we have 562.5 less or equal than straight x less or equal than 900
Hence, a minimum of 562.5 litres and a maximum of 900 litres of water be added so as to move a solution with 25% to 30% of acid content.


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